Process Capability Index - Cp vs. Cpk Visual Animation

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Thoughts about Process Capability - Cp and Cpk

Process Capability is required by TS 16949 and is part of the typical APQP process. Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification "width") to the spread of the process values, as measured by 6 process standard deviation units (the process "width").

Process
Capability Indices - A
process capability index uses both the process variability and the process
specifications to determine whether the process is "capable" We
are often required to compare the output of a stable process with the process
specifications and make a statement about how well the process meets specification.
To do this we compare the natural variability of a stable process with
the process specification limits. A capable process is one where almost
all the measurements fall inside the specification limits. There are several
statistics that can be used to measure the capability of a process: Cp,
Cpk, Cpm.Most capability indices estimates are valid only if the sample
size used is 'large enough'. Large enough is generally thought to be about
50 independent data values.

The Cp, Cpk, and Cpm statistics assume that the population of data values is normally distributed.

Capability Analysis - Process Capability Indices

Process range. First, it is customary to establish the +/- 3 sigma limits around the nominal specifications. Actually, the sigma limits should be the same as the ones used to bring the process under control using Shewhart control charts. These limits denote the range of the process (i.e., process range). If we use the +/- 3 sigma
limits then, based on the normal distribution, we can estimate that
approximately 99% of all piston rings fall within these limits.

Specification limits
LSL, USL. Usually, engineering
requirements dictate a range of acceptable values. In our example, it
may have been determined that acceptable values for the piston ring
diameters would be 74.0 +/- .02 millimeters. Thus, the lower specification
limit (LSL) for
our process is 74.0 - 0.02 = 73.98; the upper specification limit (USL) is
74.0 + 0.02 = 74.02. The difference between USL and LSL is called the specification
range. Potential capability (Cp). This
is the simplest and most straightforward indicator of process capability. It
is defined as the ratio of the specification range to the process
range; using +/- 3 sigma limits we can express this index as:
Cp = (USL-LSL)/(6*Sigma)
Put into words, this ratio expresses the proportion of the range of the
normal curve that falls within the engineering specification limits
(provided that the mean is on target, that is, that the process is centered,
see below).
Bhote (1988) reports that prior to the widespread use of
statistical quality control techniques (prior to 1980), the normal
quality of US manufacturing processes was approximately Cp =
.67. This means that the two 33/2 percent tail areas of the normal
curve fall outside specification limits. As of 1988, only about 30% of
US processes are at or below this level of quality (see Bhote, 1988, p.
51). Ideally, of course, we would like this index to be greater than 1,
that is, we would like to achieve a process capability so that no (or
almost no) items fall outside specification limits. Interestingly, in
the early 1980's the Japanese manufacturing industry adopted as their
standard Cp = 1.33! The process capability required to
manufacture high-tech products is usually even higher than this;
Minolta has established a Cp index of 2.0 as their minimum
standard (Bhote, 1988, p. 53), and as the standard for its suppliers.
Note that high process capability usually implies lower, not higher
costs, taking into account the costs due to poor quality. We will
return to this point shortly.
Capability ratio (Cr). This index is equivalent to Cp;
specifically, it is computed as 1/Cp (the inverse of Cp).
Lower/upper potential capability: Cpl, Cpu. A
major shortcoming of the Cp (and Cr) index is that it
may yield erroneous information if the process is not on target, that is, if
it is not centered.
We can express non-centering via the following quantities. First, upper
and lower potential capability indices can be computed to reflect the
deviation of the observed process mean from the LSL and USL.. Assuming +/- 3 sigma limits
as the process range, we compute:
Cpl = (Mean - LSL)/3*Sigma
and
Cpu = (USL - Mean)/3*Sigma
Obviously, if these values are not identical to each other, then the process is not centered.
Non-centering correction (K). We can correct Cp for the effects of non-centering. Specifically, we can compute:
K=abs(D - Mean)/(1/2*(USL - LSL))
where
D = (USL+LSL)/2.
This correction factor expresses the non-centering (target specification minus mean) relative to the specification range.
Demonstrated excellence (Cpk). Finally, we can adjust Cp for the effect of non-centering by computing:
Cpk = (1-k)*Cp
If the process is perfectly centered, then k is equal to zero, and Cpk is equal to Cp. However, as the process drifts from the target specification, k increases and Cpk becomes smaller than Cp.
Potential Capability II: Cpm. A recent modification (Chan, Cheng, & Spiring, 1988) to Cp is directed at adjusting the estimate of sigma for the effect of (random) non-centering. Specifically, we may compute the alternative sigma (Sigma2) as:
Sigma2 = { (xi - TS)2/(n-1)}½
where: Sigma2 is the alternative estimate of sigmaxi is the value of the i'th observation in the sampleTS is the target or nominal specificationn is the number of observations in the sample We may then use this alternative estimate of sigma to compute Cp as before; however, we will refer to the resultant index as Cpm.

Process Performance
vs. Process Capability

When monitoring a process via a quality control chart (e.g., the X-bar and R-chart;)
it is often useful to compute the capability indices for the process.
Specifically, when the data set consists of multiple samples, such as
data collected for the quality control chart, then one can compute two
different indices of variability in the data. One is the regular
standard deviation for all observations, ignoring the fact that the
data consist of multiple samples; the other is to estimate the
process's inherent variation from the within-sample variability. For
example, when plotting X-bar and R-charts one may use the common
estimator R-bar/d2 for the process sigma (e.g., see Duncan,
1974; Montgomery, 1985, 1991). Note however, that this estimator is
only valid if the process is statistically stable. For a detailed
discussion of the difference between the total process variation and
the inherent variation refer to ASQC/AIAG reference manual (ASQC/AIAG,
1991, page 80).

When the total process variability is used in the standard
capability computations, the resulting indices are usually referred to
as process performance indices (as they describe the actual performance
of the process), while indices computed from the inherent variation
(within- sample sigma) are referred to as capability indices (since
they describe the inherent capability of the process).